Operator matrices and reaction-diffusion systems

We show how the recent «matrix theory» for unbounded operator matrices can be used in order to discuss linear reaction-diffusion systems. In particular we obtain information on the existence of a dominant eigenvalue and on the asymptotic behavior of the solutions.     

A new development for the Tikhonov Theorem in nonlinear singular perturbation systems

This paper deals with the exponential stability of nonlinear perturbation systems under a new condition. A novel criterion of exponential stability of nonlinear systems is firstly given in a general form. In this criterion, a new kind of characteristic value is introduced, which makes the exponential stability measurable. Based on this criterion, a new development for the Tikhonov Theorem in nonlinear singular perturbation systems is presented.     

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.     

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results. This revised version was published online in June 2006 with corrections to the Cover Date.     

A second order perturbation expansion for small singular values

An expansion for the square of the smallest singular value of a matrix is presented. The expansion includes second order terms in the pertubation and therefore remains accurate when the smallest singular value is zero.     

Finite-difference method for singular nonlinear systems

This paper presents a method for solving nonlinear system with singular Jacobian at the solution. The convergence rate in the case of singularity deteriorates and one way to accelerate convergence is to form bordered system. A local algorithm, with finite-difference approximations, for forming and solving such system is proposed in this paper. To overcome the need that initial approximation has to be very close to the solution, we also propose a method which is a combination of descent method with finite-differences and local algorithm. Some numerical results obtained on relevant examples are presented.     

Uniform fourth order difference scheme for a singular perturbation problem

Summary The numerical solution of a nonlinear singularly perturbed two-point boundary value problem is studied. The developed method is based on Hermitian approximation of the second derivative on special discretization mesh. Numerical examples which demonstrate the effectiveness of the method are presented.This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grants JF 544, JF 799)     

Dynamics of the nonlinear stochastic heat equation with singular perturbation

The approximation in probability for a singular perturbed nonlinear stochastic heat equation is studied. First the approximation result in the sense of probability is obtained for solutions defined on any finite time interval. Furthermore it is proved that the long time behavior of the stochastic system is described by a global random attractor which is upper semi-continuous with respect to the singular perturbed parameter. This also means the long time effectivity of the approximation with probability one.     

Stability of solutions for nonlinear singular systems with delay

This work gives new notions of stability for singular systems with delay, and obtains the sufficient conditions of uniformly stability for a class of nonlinear singular systems with delay. As an application of this, an example is provided to illustrate our results.     

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.     

Initial-value technique for a class of nonlinear singular perturbation problems

An initial-value technique, which is simple to use and easy to implement, is presented for a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. It is distinguished by the following fact: The original second-order problem is replaced by an asymptotically equivalent first-order problem and is solved as an initial-value problem. Numerical experience with several examples is described.     

Limit-point criteria for semi-degenerate singular Hamiltonian differential systems with perturbation terms

Some limit-point criteria are obtained for higher-dimensional semi-degenerate singular Hamiltonian differential systems with perturbation potential terms by using M(λ)-theory. Results in this paper cover many previous results of Hartman, Levinson, Titchmarsh and Read.     

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.     

Practical Quasi-Newton algorithms for singular nonlinear systems

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained.     

一类具非线性边值条件的微分方程的奇摄动问题

研究了一类非线性三阶微分方程的奇摄动问题.运用合成展开法构造了问题的形式渐近解,并运用不动点原理证明了原问题解的存在性及所得形式渐近解的一致有效性.     

Algebraic perturbation methods for the solution of singular linear systems

A singular matrix A is perturbed algebraically to obtain a nonsingular matrix B. Particular solutions of Ax=b can be found as unique solutions of Bx=d, where d is an algebraic perturbation of b. More specially, null vectors and generalized null vectors of A can be found as unique solutions of linear systems. It is shown also that B^?1AB^?1 is a generalized inverse of A.     

Mathematical modeling of fractional reaction-diffusion systems with different order time derivatives

The linear stability analysis is studied for a two-component fractional reaction-diffusion system with different derivative indices. Two different cases are considered: when the activator index is larger than the inhibitor one and when the inhibitor variable index is larger than the activator one. The general analysis is confirmed by computer simulation of the system with cubic nonlinearity. It is shown that systems with a higher activator variable index lead to a much more complicated space-time dynamics.